module Language.Traces where

open import Language.Base
open import Language.Semantics using (Env; _,_⇒ᵇˢ_)
open import Language.Graphs

open import Data.Product using (_,_)
open import Data.List.Membership.Propositional using (_∈_)

module _ {g : Graph} where
    open Graph g using (Index; edges; inputs; outputs)

    data Trace : Index → Index → Env → Env → Set where
        Trace-single : ∀ {ρ₁ ρ₂ : Env} {idx : Index} →
                       ρ₁ , (g [ idx ]) ⇒ᵇˢ ρ₂ → Trace idx idx ρ₁ ρ₂
        Trace-edge : ∀ {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : Index} →
                     ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) ∈ edges →
                     Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃

    infixr 5 _++⟨_⟩_
    _++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} →
           Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) ∈ edges →
           Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃
    _++⟨_⟩_ (Trace-single ρ₁⇒ρ₂) idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₂→idx₃ tr
    _++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr)

    record EndToEndTrace (ρ₁ ρ₂ : Env) : Set where
        constructor MkEndToEndTrace
        field
            idx₁ : Index
            idx₁∈inputs : idx₁ ∈ inputs

            idx₂ : Index
            idx₂∈outputs : idx₂ ∈ outputs

            trace : Trace idx₁ idx₂ ρ₁ ρ₂